In this paper we study a complete quadrangle in the Euclidean plane that has a rectangular hyperbola circumscribed to it. Hereby, the approach is based on the rectangular coordinates and we prove the following main result: Let ABCD be a complete quadrangle and l a , l b , l c , l d mutually parallel lines through the circumcenters of BCD , ACD , ABD , ABC , respectively. Orthopoles of the lines l a , l b , l c , l d with respect to the triangles BCD , ACD , ABD , ABC lie on a line which passes through the center of the rectangular hyperbola H circumscribed to ABCD , and it is antiparallel to the given lines with respect to the axes of the hyperbola H .